[Paper Review] Weak convergence of regular Dirichlet subspaces

Liping Li, Toshihiro Uemura+1 more|arXiv (Cornell University)|Sep 6, 2015

advanced mathematical theories6 references3 citations

TL;DR

This paper establishes weak convergence of diffusion processes associated with regular Dirichlet subspaces of a fixed 1D diffusion, under monotonicity of their characteristic sets—either decreasing or increasing. The key result shows that if the characteristic sets converge in the $d\mathtt{s}$-a.e. sense, then the associated diffusion processes converge weakly, extending prior Mosco convergence results to pathwise convergence.

ABSTRACT

In this paper we shall prove the weak convergence of the associated diffusion processes of regular subspaces with monotone characteristic sets for a fixed Dirichlet form. More precisely, given a fixed 1-dimensional diffusion process and a sequence of its regular subspaces, if the characteristic sets of regular subspaces are decreasing or increasing, then their associated diffusion processes are weakly convergent to another diffusion process. This is an extended result of [13].

Motivation & Objective

• To extend previous Mosco convergence results on regular Dirichlet subspaces to weak convergence of associated diffusion processes.
• To investigate the convergence behavior of diffusion processes when the characteristic sets of regular subspaces are monotonic (increasing or decreasing) in the $d\mathtt{s}$-a.e. sense.
• To establish sufficient conditions under which the associated diffusion processes of regular subspaces converge weakly to another diffusion process.
• To analyze the relationship between the limiting behavior of characteristic sets and the convergence of Dirichlet forms and their associated processes.

Proposed method

• The paper studies regular Dirichlet subspaces of a fixed 1D diffusion via their associated scaling functions $\mathtt{s}_n$ and characteristic sets $G_n = \{x : d\mathtt{s}_n/d\mathtt{s} = 1\}$.
• It uses the Lyons-Zheng decomposition to analyze sample path continuity and tightness of the diffusion processes $\mathbf{X}^n$.
• Tightness is established by bounding the energy measure $\mu_{\langle f\rangle}$ of the martingale part, which equals $\overset{\circ}{\mathtt{s}}(dx)$, the Lebesgue measure on $I$.
• The proof relies on time reversal and Brownian motion coupling to control the modulus of continuity of $Z_t = X_t^t$.
• It applies the condition $d\overset{\circ}{\mathtt{s}}/dm \leq C$ to bound the quadratic variation and ensure uniform tightness across $n$.
• The convergence of $\mathbf{X}^n$ is shown via tightness and finite-dimensional distribution convergence, leveraging the weak convergence criteria.

Experimental results

Research questions

• RQ1 Under what conditions does the sequence of diffusion processes $\mathbf{X}^n$ associated with regular Dirichlet subspaces converge weakly?
• RQ2 How does the monotonicity of characteristic sets $G_n$ (increasing or decreasing) affect the weak convergence of $\mathbf{X}^n$?
• RQ3 Can Mosco convergence of Dirichlet forms be strengthened to weak convergence of associated diffusion processes?
• RQ4 What happens to the limiting process when the characteristic sets converge to a non-characteristic set or to the full interval?

Key findings

• If the characteristic sets $G_n$ are monotonic (either $G_n \downarrow G$ or $G_n \uparrow G$) in the $d\mathtt{s}$-a.e. sense, then the associated diffusion processes $\mathbf{X}^n$ converge weakly to a diffusion process $\mathbf{X}$.
• The weak convergence holds under mild conditions, including boundedness of $d\overset{\circ}{\mathtt{s}}/dm$, which ensures uniform tightness of the processes.
• The proof establishes tightness by showing that the modulus of continuity of $Z_t = X_t^t$ vanishes uniformly as $\delta \to 0$, using Brownian motion coupling and time reversal.
• The energy measure of the martingale part in the Lyons-Zheng decomposition equals $\overset{\circ}{\mathtt{s}}(dx)$, which is crucial for bounding the quadratic variation.
• The result extends prior Mosco convergence results from [13] to weak convergence of processes, even in cases where the limiting form may be degenerate (e.g., zero form, as in Example 3.10).
• The convergence is robust: the limiting process $\mathbf{X}$ corresponds to a Dirichlet form with scaling function $\mathtt{s}_\infty$, whose characteristic set is the limit of $G_n$.